Asymptotic lattices and their integrable reductions I: the Bianchi and the Fubini-Ragazzi lattices

Abstract

We review recent results on asymptotic lattices and their integrable reductions. We present the theory of general asymptotic lattices in R^3 together with the corresponding theory of their Darboux-type transformations. Then we study the discrete analogues of the Bianchi surfaces and their transformations. Finally, we present the corresponding theory of the discrete analogues of the isothermally-asymptotic (Fubuni-Ragazzi) nets.

arXiv comment16 pages, 2 figures, uses iopart style

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Cited by in corpus (10)

• Isothermic surfaces in sphere geometries as Moutard nets
• Non-commutative lattice modified Gel'fand-Dikii systems
• Discretization of asymptotic line parametrizations using hyperboloid patches
• Geometric discretization of the Koenigs nets
• Integrable Systems and Discrete Geometry
• Generalized isothermic lattices
• Geometric discretization of the Bianchi system
• On Weingarten transformations of hyperbolic nets
• Hirota equation and the quantum plane
• Bäcklund transformations as integrable discretization. The geometric approach